Monday, 16 June 2014

limits - limlimitsnrightarrow+inftyfracln(1+n+n3)3ln(n)n(1cos(1/n2))



I want to solve this limit:



limn+ln(1+n+n3)3ln(n)n(1cos(1/n2))



I have proved that limn+ln(1+n+n3)3ln(n)n=0 and limn+1(1cos(1/n2))= but I have indeterminate form. How can I solve that?


Answer



A long version:
limn+ln(1+n+n3)3ln(n)n(1cos(1/n2))=limn+ln(1+n+n3)ln(n3)n1n4(1cos2(1/n2))1n4(1+cos(1/n2))=limn+ln(1+n+n3)ln(n3)1n3sin2(1/n2)1n4(1+cos(1/n2))=2limn+ln(1+n+n3)ln(n3)1n3=2limn+n3ln(1+n+n3n3)=2limn+ln(1+1+nn3)n3=2limn+ln(1+1+nn3)n3n+1(n+1)=2limn+(n+1)ln(1+1+nn3)n3n+1=2lnelimn+(n+1)+







On the 2nd line
limn+sin2(1/n2)1n4=limn+sin(1/n2)1n2sin(1/n2)1n2=1 from
limx0sinxx=1


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